Integrand size = 22, antiderivative size = 66 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^2 \, dx=b \sqrt {c+\frac {d}{x^2}} x+\frac {a \left (c+\frac {d}{x^2}\right )^{3/2} x^3}{3 c}-b \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right ) \] Output:
b*(c+d/x^2)^(1/2)*x+1/3*a*(c+d/x^2)^(3/2)*x^3/c-b*d^(1/2)*arctanh(d^(1/2)/ (c+d/x^2)^(1/2)/x)
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.03 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^2 \, dx=\frac {1}{3} \sqrt {c+\frac {d}{x^2}} x \left (3 b+a \left (\frac {d}{c}+x^2\right )-\frac {3 b \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+c x^2}}{\sqrt {d}}\right )}{\sqrt {d+c x^2}}\right ) \] Input:
Integrate[(a + b/x^2)*Sqrt[c + d/x^2]*x^2,x]
Output:
(Sqrt[c + d/x^2]*x*(3*b + a*(d/c + x^2) - (3*b*Sqrt[d]*ArcTanh[Sqrt[d + c* x^2]/Sqrt[d]])/Sqrt[d + c*x^2]))/3
Time = 0.33 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {953, 773, 247, 224, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^2 \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} \, dx\) |
\(\Big \downarrow \) 953 |
\(\displaystyle b \int \sqrt {c+\frac {d}{x^2}}dx+\frac {a x^3 \left (c+\frac {d}{x^2}\right )^{3/2}}{3 c}\) |
\(\Big \downarrow \) 773 |
\(\displaystyle \frac {a x^3 \left (c+\frac {d}{x^2}\right )^{3/2}}{3 c}-b \int \sqrt {c+\frac {d}{x^2}} x^2d\frac {1}{x}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {a x^3 \left (c+\frac {d}{x^2}\right )^{3/2}}{3 c}-b \left (d \int \frac {1}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}-x \sqrt {c+\frac {d}{x^2}}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {a x^3 \left (c+\frac {d}{x^2}\right )^{3/2}}{3 c}-b \left (d \int \frac {1}{1-\frac {d}{x^2}}d\frac {1}{\sqrt {c+\frac {d}{x^2}} x}-x \sqrt {c+\frac {d}{x^2}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {a x^3 \left (c+\frac {d}{x^2}\right )^{3/2}}{3 c}-b \left (\sqrt {d} \text {arctanh}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )-x \sqrt {c+\frac {d}{x^2}}\right )\) |
Input:
Int[(a + b/x^2)*Sqrt[c + d/x^2]*x^2,x]
Output:
(a*(c + d/x^2)^(3/2)*x^3)/(3*c) - b*(-(Sqrt[c + d/x^2]*x) + Sqrt[d]*ArcTan h[Sqrt[d]/(Sqrt[c + d/x^2]*x)])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^ 2, x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && !IntegerQ[p]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[d/e^n Int[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && G tQ[m + n, -1]))
Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.26
method | result | size |
default | \(-\frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x \left (3 \sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) b c -a \left (c \,x^{2}+d \right )^{\frac {3}{2}}-3 \sqrt {c \,x^{2}+d}\, b c \right )}{3 \sqrt {c \,x^{2}+d}\, c}\) | \(83\) |
Input:
int((a+b/x^2)*(c+d/x^2)^(1/2)*x^2,x,method=_RETURNVERBOSE)
Output:
-1/3*((c*x^2+d)/x^2)^(1/2)*x*(3*d^(1/2)*ln(2*(d^(1/2)*(c*x^2+d)^(1/2)+d)/x )*b*c-a*(c*x^2+d)^(3/2)-3*(c*x^2+d)^(1/2)*b*c)/(c*x^2+d)^(1/2)/c
Time = 0.11 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.27 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^2 \, dx=\left [\frac {3 \, b c \sqrt {d} \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + 2 \, {\left (a c x^{3} + {\left (3 \, b c + a d\right )} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{6 \, c}, \frac {3 \, b c \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{d}\right ) + {\left (a c x^{3} + {\left (3 \, b c + a d\right )} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{3 \, c}\right ] \] Input:
integrate((a+b/x^2)*(c+d/x^2)^(1/2)*x^2,x, algorithm="fricas")
Output:
[1/6*(3*b*c*sqrt(d)*log(-(c*x^2 - 2*sqrt(d)*x*sqrt((c*x^2 + d)/x^2) + 2*d) /x^2) + 2*(a*c*x^3 + (3*b*c + a*d)*x)*sqrt((c*x^2 + d)/x^2))/c, 1/3*(3*b*c *sqrt(-d)*arctan(sqrt(-d)*x*sqrt((c*x^2 + d)/x^2)/d) + (a*c*x^3 + (3*b*c + a*d)*x)*sqrt((c*x^2 + d)/x^2))/c]
Time = 3.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.62 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^2 \, dx=\frac {a \sqrt {d} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{3} + \frac {a d^{\frac {3}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{3 c} + \frac {b \sqrt {c} x}{\sqrt {1 + \frac {d}{c x^{2}}}} - b \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )} + \frac {b d}{\sqrt {c} x \sqrt {1 + \frac {d}{c x^{2}}}} \] Input:
integrate((a+b/x**2)*(c+d/x**2)**(1/2)*x**2,x)
Output:
a*sqrt(d)*x**2*sqrt(c*x**2/d + 1)/3 + a*d**(3/2)*sqrt(c*x**2/d + 1)/(3*c) + b*sqrt(c)*x/sqrt(1 + d/(c*x**2)) - b*sqrt(d)*asinh(sqrt(d)/(sqrt(c)*x)) + b*d/(sqrt(c)*x*sqrt(1 + d/(c*x**2)))
Time = 0.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.14 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^2 \, dx=\frac {a {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} x^{3}}{3 \, c} + \frac {1}{2} \, {\left (2 \, \sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )\right )} b \] Input:
integrate((a+b/x^2)*(c+d/x^2)^(1/2)*x^2,x, algorithm="maxima")
Output:
1/3*a*(c + d/x^2)^(3/2)*x^3/c + 1/2*(2*sqrt(c + d/x^2)*x + sqrt(d)*log((sq rt(c + d/x^2)*x - sqrt(d))/(sqrt(c + d/x^2)*x + sqrt(d))))*b
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (54) = 108\).
Time = 0.13 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.76 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^2 \, dx=\frac {b d \arctan \left (\frac {\sqrt {c x^{2} + d}}{\sqrt {-d}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-d}} - \frac {{\left (3 \, b c d \arctan \left (\frac {\sqrt {d}}{\sqrt {-d}}\right ) + 3 \, b c \sqrt {-d} \sqrt {d} + a \sqrt {-d} d^{\frac {3}{2}}\right )} \mathrm {sgn}\left (x\right )}{3 \, c \sqrt {-d}} + \frac {{\left (c x^{2} + d\right )}^{\frac {3}{2}} a c^{2} \mathrm {sgn}\left (x\right ) + 3 \, \sqrt {c x^{2} + d} b c^{3} \mathrm {sgn}\left (x\right )}{3 \, c^{3}} \] Input:
integrate((a+b/x^2)*(c+d/x^2)^(1/2)*x^2,x, algorithm="giac")
Output:
b*d*arctan(sqrt(c*x^2 + d)/sqrt(-d))*sgn(x)/sqrt(-d) - 1/3*(3*b*c*d*arctan (sqrt(d)/sqrt(-d)) + 3*b*c*sqrt(-d)*sqrt(d) + a*sqrt(-d)*d^(3/2))*sgn(x)/( c*sqrt(-d)) + 1/3*((c*x^2 + d)^(3/2)*a*c^2*sgn(x) + 3*sqrt(c*x^2 + d)*b*c^ 3*sgn(x))/c^3
Time = 3.73 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.21 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^2 \, dx=b\,x\,\sqrt {c+\frac {d}{x^2}}+\frac {a\,x\,\sqrt {c+\frac {d}{x^2}}\,\left (c\,x^2+d\right )}{3\,c}+\frac {b\,\sqrt {d}\,\mathrm {asin}\left (\frac {\sqrt {d}\,1{}\mathrm {i}}{\sqrt {c}\,x}\right )\,\sqrt {c+\frac {d}{x^2}}\,1{}\mathrm {i}}{\sqrt {c}\,\sqrt {\frac {d}{c\,x^2}+1}} \] Input:
int(x^2*(a + b/x^2)*(c + d/x^2)^(1/2),x)
Output:
b*x*(c + d/x^2)^(1/2) + (a*x*(c + d/x^2)^(1/2)*(d + c*x^2))/(3*c) + (b*d^( 1/2)*asin((d^(1/2)*1i)/(c^(1/2)*x))*(c + d/x^2)^(1/2)*1i)/(c^(1/2)*(d/(c*x ^2) + 1)^(1/2))
Time = 0.19 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.50 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^2 \, dx=\frac {\sqrt {c \,x^{2}+d}\, a c \,x^{2}+\sqrt {c \,x^{2}+d}\, a d +3 \sqrt {c \,x^{2}+d}\, b c +3 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x -\sqrt {d}}{\sqrt {d}}\right ) b c -3 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x +\sqrt {d}}{\sqrt {d}}\right ) b c}{3 c} \] Input:
int((a+b/x^2)*(c+d/x^2)^(1/2)*x^2,x)
Output:
(sqrt(c*x**2 + d)*a*c*x**2 + sqrt(c*x**2 + d)*a*d + 3*sqrt(c*x**2 + d)*b*c + 3*sqrt(d)*log((sqrt(c*x**2 + d) + sqrt(c)*x - sqrt(d))/sqrt(d))*b*c - 3 *sqrt(d)*log((sqrt(c*x**2 + d) + sqrt(c)*x + sqrt(d))/sqrt(d))*b*c)/(3*c)